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Math Expert V
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Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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Question Stats: 79% (01:02) correct 21% (01:13) wrong based on 418 sessions

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Which of the following inequalities is equivalent to −4 < x < 8?

A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above

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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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Bunuel wrote:
Which of the following inequalities is equivalent to −4 < x < 8?

A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above

|x - a| = b

x is a distance 'b' away from 'a' on either side on the number line.

-4 < x < 8
Mid point of -4 and 8 is 2. Both -4 and 8 are 6 away from 2.
So the absolute value form will be |x - 2| < 6

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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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We know that |x| < a means -a < x < a, where Sum of lower limit of x (i.e -a) and the upper limit of x (i.e a), is 0

Given is, -4 < x < 8, let's say by adding y to this inequality we will get into the above format

-4+y < x+y < 8+y

Now, to move this into the mod format, we need to have (-4+y) + (8+y) = 0 => y = -2

Thus, -6< x-2 < 6 => |x-2| < 6.

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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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Bunuel wrote:
Which of the following inequalities is equivalent to −4 < x < 8?

A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above

Given −4 < x < 8,
Now try to get into the mod form,
subtract two from both sides of the equation, (graphically the line remains the same)

-4 -2 < x-2 < 8-2 => -6 < x-2 < 6 and this is same as | x-2 | < 6, which is D

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Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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The inequality $$|x-a|<b$$ is equivalent to inequality $$a- b < x< a+b$$ in which a is average of a+b & a-b, and b is mid distance from $$a-b$$and $$a+b.$$

So $$a = {(a+b) + (a-b)}/2$$
$$b= {(a+b) - (a-b)}/2$$

Let's put these values in original equation.

$$a= (-4+8)/2 =2$$

$$b={8-(-4)}/2 = 6$$

$$|x-2|<6$$

Similar question to practice: https://gmatclub.com/forum/which-of-the ... 59334.html
Thanks to MathRevolution for detail explanation.
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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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Bunuel wrote:
Which of the following inequalities is equivalent to −4 < x < 8?

A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above

We see that the first 4 given choices are all in the form of |x - a| < b (for example, |x + 2| < 6 can be considered as |x - (-2)| < 6).

Recall that |x - a| < b (where b is positive) means -b < x - a < b, which becomes a - b < x < a + b. So we want to find a and b such that a - b = -4 and a + b = 8. Adding these two equations, we have: 2a = 4 → a = 2. Substituting 2 for a in a + b = 8, we have b = 6. Therefore, |x - 2| < 6 is the correct inequality.

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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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Question)
Which of the following inequalities is equivalent to −4 < x < 8?
A. |x - 1| < 7
B. |x + 2| < 6
C. |x + 3| < 5
D. |x - 2| < 6
E. None of the above

Solution:

We need to form an inequality with absolute value for the inequality −4 < x < 8
This implies that x will take all values between −4 and 8

The mean of −4 and 8 is 2
Also, the distance of the mean (i.e. 2) from −4 or from 8 is 6
Thus: −4 = (2 − 6) and 8 = (2 + 6)

We know that |x − p | = q implies that the distance of x from p is q units
This results in the values (p − q) and (p + q) for x

Thus, comparing, if we have: |x − 2| = 6, we would get x = 2 − 6 = −4 or x = 2 + 6 = 8
Since we need all the values between −4 and 8 (i.e. we should NOT exceed −4 or 8), we should therefore have:
|x − 2| < 6

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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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In this question on inequalities and absolute values, the concept that is being tested is as follows:

If |x-a| < k, then the range of x that satisfies the inequality is given by a-k < x < a+k.

Comparing this with the inequality given in the question, we can say a-k = -4 and a+k = 8. This means a = 2 and k = 6.

This means that -4<x<8 is the range that satisfies the inequality |x-2| < 6. The correct answer option is D.

Hope that helps!
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Re: Which of the following inequalities is equivalent to −4 < x < 8?  [#permalink]

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This case is simple use the pattern of absolute values |x − a| = b and then x = a+b and aslo x = a-b

In our case we have a-b = -4 and a+b = 8 so we can try each example to find the correct answer. which is |x − 2| = 6
2 + 6 = 8 and 2-6 = -4 Re: Which of the following inequalities is equivalent to −4 < x < 8?   [#permalink] 13 Feb 2020, 07:52
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